They could act any way, really. It's not the class or the god they worship that makes a person, so much as how you decide they aught to behave.
However, looking at the Black Butterfly's information, there are a few good places to take some inspiration from.
A big part of her worship is anonymous donation and charity, to the point that if a worshiper is caught performing the act, they're expected to pray blindfolded for an hour then tithe to a local good-aligned temple (likely with preference to Desna).
There's of course her anti Dark Tapestry angle, so a worshiper of hers would probably be well schooled in identifying agents and aberrations belonging to the Dark Tapestry, and would probably seek to eradicate them wherever they're found.
There's also a large aspect of freedom and liberation associated with her worship, as well as travel and exploration, very similar to Desna.
It's pretty likely a worshiper of the Black Butterfly would even be mistaken for an offshoot Desnan priest to the typical yokel.
Ultimately if I were to run an inquisitor of the Black Butterfly, I'd probably have them behave very similarly to a Desnan priest, but up the mysteriousness and enigmatic nature. They would go out of their way to do good, but never seek praise for it - that seems pretty anathema to how the Black Butterfly would expect a worshiper to act.